What are mono-variable and multi-variable neural networks?
What are mono-variable and multi-variable neural networks?

I am not sure about this, because most (if not all useful) neural networks are multivariable neural networks (i.e. they contain multiple parameters). Even the perceptron usually contains more than one parameter, so that terminology isn't clear even to me. Maybe they are referring to the number of inputs (sometimes called variables), but I don't see why this distinction in this context would make sense.

What are static and dynamic neural networks?

To answer this question, I will first quote an excerpt from this document (written in Spanish) to provide some context to Spanish speakers (I am not a Spanish speaker, but I understand 99% of it).

Un primer intento de clasificación puede separ estáticos y dinámicos o recurrentes (fig.2.1)
Los modelos estáticos realizan un mapeo entre entrada y salida. Despreciando el tiempo de procesamiento interno, la salida se obtiene en forma inmediata en función de la entrada, no existe memoria ni dinámica de estados en el sistema neuronal.
Por el contrario los sistemas recurrentes si la poseen, son sistemas realimentados que ante un estimulo de entrada evolucionan hasta converger a una salida estable.
Casos tipicos de ambos sistemas son el Perceptrón (Rosemblatt, 1960a) (de una o múltiples capas) y la memoria asociativa de Hopfield, respectivamente
(Tank, 1987).

So, in this document, the word &quot;dynamic&quot; and &quot;recurrent&quot; are being used interchangeably. An example of a static (i.e. non-recurrent) neural network is the perceptron. An example of recurrent (or dynamic) neural network is the Hopfield network.
Anyway, I recommend you contact the author of that article to ask for clarification (especially, about the mono-variable NNs)!